Donald J. Trump still has two rivals left for the Republican presidential nomination, but the talk in political circles has turned to what kind of running mate might strengthen his White House bid if he secures the nomination – and who might (or might not) be interested in joining the ticket.

Here are some of those names – according to Republican strategists, advisers and allies of potential picks, and some of the contenders themselves – and how they may be leaning.

Ben Carson

Who He Is: Retired neurosurgeon, former Republican presidential candidate (2016).

Why Trump Might Pick Him: He’s popular with social conservatives and evangelicals, and he impressed Mr. Trump when he put their campaign skirmishes behind him to endorse him in March. Mr. Carson can be a compelling speaker when he draws from his experiences as a doctor and from his faith, and as an African-American, he would bring diversity to the ticket.

What Carson Might Say: He told The Times that he would consider being Mr. Trump’s running mate “if it was really felt that I would add enormously and bring something that other people wouldn’t bring.” But Mr. Carson also acknowledged that he didn’t have the “government experience” that he thinks Mr. Trump needs on the ticket.

Chris Christie

Who He Is: Governor of New Jersey since 2010, former Republican presidential candidate (2016).

Why Trump Might Pick Him: Mr. Christie has been one of Mr. Trump’s most loyal and full-throated supporters since he dropped out of the race and endorsed Mr. Trump, with whom he has a long friendship. Mr. Christie has expertise in law enforcement and criminal justice issues, experience running a government and dealing with immigration issues, all of which could be valuable qualities. Mr. Trump may also appreciate that Mr. Christie is a fellow executive and plain-speaking leader.

What Christie Might Say: Mr. Christie is set to leave office in early 2018, and he has expressed openness to considering another role in government. He is clearly interested in the presidency, having run an aggressive campaign, and, at 53, he could easily run again in 2024 after two Trump terms. But he may be more likely to be looking at a cabinet position or other job in the administration, if Mr. Trump decides that Mr. Christie doesn’t bring enough to the ticket or is too tainted by controversies like the George Washington Bridge scandal.

Mary Fallin

Who She Is: Governor of Oklahoma since 2011, former legislator and lieutenant governor in Oklahoma and House member.

Why Trump Might Pick Her: Ms. Fallin, 61, has long experience in government, with a focus on national security – a relatively new area for Mr. Trump. She might help him with female voters, particularly with his goal of winning a majority of the vote from white women, which Republican nominees have done since 1996. When a political commentator recently suggested Ms. Fallin as a running mate, Mr. Trump tweeted that it was “great” advice.
What Fallin Might Say: While Ms. Fallin has said that her “first and foremost goal” is serving Oklahoma as governor, allies say she is intrigued by the idea of joining a Trump ticket. They add that she is not quite sure what to make of him, because she does not know him well, but is open to being vetted and considered.

Newt Gingrich

Who He Is: Former House speaker, former Republican presidential candidate (2012).

Why Trump Might Pick Him: Mr. Gingrich made his name taking on President Bill Clinton during the “Republican Revolution” of 1994 — the sort of fiery, insurgent operation that Mr. Trump seems to appreciate. He’s also just idiosyncratic enough to handle Mr. Trump’s strong personality, and has generally offered a series of positive comments about him.

What Gingrich Might Say: He said “it would be very hard for a patriotic citizen to say no,” and very few people pass up the chance to be heartbeat away from the presidency. The spot would give Mr. Gingrich, who is 72, what would probably be a final shot at elected office.
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Nikki R. Haley

Who She Is: Governor of South Carolina since 2011.

Why Trump Might Pick Her: Ms. Haley is widely regarded in national Republican circles and has been a popular governor, including among some Democrats. She has aggressively recruited companies to open offices and plants in South Carolina as part of her overall focus on job creation, a high priority for Mr. Trump. Ms. Haley, 44, is also one of the most prominent women in American politics, and Mr. Trump is facing a likely female opponent in the general election and sluggish support among female voters.
What Haley Might Say: Ms. Haley, who endorsed Senator Marco Rubio ahead of the South Carolina primary, has described Mr. Trump’s politics as divisive and blasted him for not swiftly repudiating support from a former Ku Klux Klan leader. Mr. Trump has hit back, writing on Twitter in March that South Carolinians were “embarrassed” by Ms. Haley. She has told allies that she isn’t interested in the vice presidency, but a reconciliation between her and Mr. Trump is surely possible if he becomes the leader of the party.

John Kasich

Who He Is: Governor of Ohio since 2011, current Republican presidential candidate, a former nine-term House member who was chairman of the House Budget Committee.

Why Trump might pick him: Mr. Kasich knows his way around Washington and governing — qualifications Mr. Trump has said he values. A sitting governor of a critical swing state, he would do well in a general election match-up against Hillary Clinton, polls show. He is also well-liked by Washington establishment types, which could help if Mr. Trump faces a brokered convention.
What Kasich Might Say: “Never. No chance,” his spokesman said of the possibility. Mr. Kasich has campaigned as a happy warrior and a moderate, a sort self-styled conscience of the establishment wing of the party — and Mr. Trump in many ways represents the exact opposite. But that No. 2 slot might look more appealing if it looks like his campaign is ending.

Rob Portman

Who He Is: Senator from Ohio since 2011, served in the House for 12 years, former United States trade representative and director of the Office of Management and Budget under President George W. Bush.

Why Trump Might Pick Him: Mr. Portman brings the Washington résumé and know-how Mr. Trump desperately needs. A sitting senator from a swing state, he is a perennial favorite on vice-presidential short lists.
What Portman Might Say: As he said with a laugh on Thursday when asked, “I’m focused on my own race.” Mr. Portman is facing a tough re-election fight in Ohio, and aligning himself with Mr. Trump could hurt his shot at another term.

​Joe Scarborough

Who He Is: Host of “Morning Joe” on MSNBC and a former Republican congressman from Florida who served on the House Armed Services Committee.

Why Trump Might Pick Him: Mr. Scarborough, whose morning show guests are a who’s-who of Acela corridor insiders, offers Mr. Trump a certain Washington savvy. But he’s also the sort of unconventional, celebrity choice Mr. Trump just might gravitate toward. And the two men enjoy a friendly relationship and easy rapport on “Morning Joe.”

What Scarborough Might Say: He would have to give up his MSNBC show. But the No. 2 slot offers plenty of upside for Mr. Scarborough, who has long flirted with a return to politics. Even a loss would catapult him to the next level of fame and name recognition — either increasing ratings for his show, or paving his return to politics.

Tim Scott

Who He Is: Senator from South Carolina since 2013.

Why Trump Might Pick Him: A fiscal and social conservative who is admired by different segments of the party, Mr. Scott is also one of the few black Republicans holding a major office. He might increase Mr. Trump’s appeal with some black voters, while also energizing Tea Party conservatives and evangelicals to turn out to vote. Mr. Scott, 50, has also been a small-business man and has some relationships with members of Congress that could be be useful.
What Scott Might Say: Still in his first term, he is focused on his re-election race this year, and he isn’t particularly inclined toward Mr. Trump, endorsing Mr. Rubio in the South Carolina primary. The two men do share views about reducing the size of government, but Mr. Scott has faulted Mr. Trump for not quickly disavowing support from the former Klan leader. He said in an interview on Thursday that he was not interested in being considered.

Jeff Sessions

Who He Is: Senator from Alabama since 1997, the first (and still only) sitting senator to endorse Mr. Trump, and a senior adviser to the campaign.

Why Trump Might Pick Him: Mr. Sessions, who sometimes appears with Mr. Trump on the campaign trail as an opening act, is fiercely aligned with Mr. Trump on illegal immigration. And he sits on several key Senate committees — Judiciary, Armed Services, Budget — giving him an insider’s perspective on how Washington works.

What Sessions Might Say: Though he demurred when asked if he would serve as Mr. Trump’s vice president, he did say he would undergo the vetting process if asked. An unabashed Trump supporter, it’s hard to imagine him saying no.